Gen. Math. Notes, Vol. 2, No. 1, January 2011, pp. 18-23
c
ISSN 2219-7184; Copyright °ICSRS
Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
A Two-Sided Multiplication Operator Norm
N. B. Okelo
School of Biological and Physical Sciences
Bondo University College, Box 210, Bondo, Kenya
bnyaare@yahoo.com
J. O. Agure
Department of Mathematics and Applied Statistics
Maseno University, Box 333, Maseno, Kenya
johnagure@yahoo.com
(Received 17.11.2010, Accepted 29.11.2010)
Abstract
Let A be a C ∗ -algebra and define an elementary operator Ta,b : A →
A by Ta,b (x) = Σni=1 ai xbi , ∀ x ∈ A where ai and bi are fixed in A
or multiplier algebra M (A) of A. Here, we determine the norm of a
two-sided multiplication operator.
Keywords: Two-sided Multiplication Operator, Elementary Operator, Norm.
Mathematics Subject Classification: Primary 47B47; Secondary 47A30.
1
Introduction
Let H be a complex Hilbert space and B(H) the algebra of all bounded linear
operators on H. Then T : B(H) → B(H) is an elementary operator if T has
a representation Ta,b (x) = Σni=1 ai xbi , ∀ x ∈ B(H), where ai and bi are fixed
in B(H). Some examples of elementary operators are the left multiplication
La (x) = ax; the right multiplication Rb (x) = xb; the generalized derivation
δa,b = La − Rb ; the inner derivation, the two-sided multiplication operator
A two-sided multiplication operator norm
19
Ma,b = La Rb and the Jordan elementary operator Ua,b = Ma,b + Mb,a . Determining the lower estimate of the norm of elementary operators has attracted
a lot of interest from many mathematicians (see [1-5, 7-18]). Clearly, every
elementary operator is bounded. For the lower estimates of the norms, there
have been several results obtained by different mathematicians. For example,
Mathieu [6] proved that for a prime C*- algebra A, kUa,b |Ak ≥ 32 kakkbk, Cabr1
era and Rodriguez [4] proved that for JB* algebras, kUa,b |Ak ≥ 20412
kakkbk,
while Stacho and Zalar [12] obtained results for standard
operator
algebras
√
on Hilbert spaces i.e. they showed that kUa,b |Ak ≥ 2( 2 − 1)kakkbk. Recently, Timoney [15, 16] demonstrated that kUa,b |Ak ≥ kakkbk. He [18] also
gave a formula for the norm of an elementary operator on a C*-algebra using
the notion of matrix valued numerical ranges and a kind of tracial geometric
mean.
Theorem 1.1. For a = [a1 , ..., an ] ∈ B(H)n (a row matrix of operators ai ∈
B(H)), b = [b1 , ..., bn ] ∈ B(H)n (a column matrix of operators bi ∈ B(H)) and
Ta,b (x) = Σni=1 ai xbi , ∀ x ∈ B(H), an elementary operator, we have
kT k = sup{tgm(Q(a∗ , ξ), Q(b, η)) : ξ, η ∈ H, kξk = 1, kηk = 1}.
For proof, see [18, Theorem 1.4].
Interestingly, for Calkin algebras, it has been easy to calculate the norms
of elementary operators as shown by Mathieu [7]. Considering a two-sided
multiplication operator Ma,b , it has been shown in [2], the necessary and sufficient conditions for any pair of operators a, b ∈ B(H) to satisfy the equation
kI + Ma,b k = 1 + kakkbk.
Definition 1.2. Let T ∈ B(H). The maximal numerical range of T is defined
by W0 (T ) = {λ : hT xn , xn i → λ, where kxn k = 1 and kT xn k → kT k} and the
normalized maximal numerical range is given by
½
W0 ( kTT k ), if T 6= 0,
WN (T ) =
0,
if T= 0.
The set W0 (T ) is nonempty, closed, convex and contained in the closure of the
numerical range, see [14].
Theorem 1.3. For a, b ∈ B(H) the following are equivalent:
(1) kI + Ma,b k = 1 + kakkbk,
(2) WN (a∗ ) ∩ WN (b) 6= Ø.
See [2] for proof.
Conjecture 1.4. Let A be a standard operator subalgebra of B(H). The
estimate of M, such that kMa,b xk = kakkbk holds for every a, b ∈ A.
20
N. B. Okelo and J. O. Agure
This conjecture was verified in the following cases :
(i) for a, b ∈ B(H) such that inf λ∈C ka + λbk = kak or inf λ∈C kb + λak = kbk,
(ii) in the Jordan algebra of symmetric operators. See [1, 13].
Nyamwala and Agure [8] used the spectral resolution theorem to calculate
the norm of an elementary operator induced by normal operators in a finite
dimensional Hilbert space. They gave the following result.
Theorem 1.5. Let Ta,b : B(H) → B(H) be an elementary operator defined
P
by Ta,b (x) = ki=1 ai xbi where ai and bi are normal operators and H a finite
m−dimensional Hilbert space then
k X
m
X
1
kT k = ( (
| αi,j |2 | βi,j |2 )) 2
j=1 j=1
where αi,j and βi,j are distinct eigenvalues of ai and bi respectively.
A specific example in [ 8, Example 2.3 ] shows that kT k = 2. In the next
section, we determine the norm of a two-sided multiplication operator.
2
Two-sided Multiplication Operator Norm
In this section we concentrate on a complex Hilbert space over the field K. We
show that for a two-sided multiplication operator M, kMa,b xk = kakkbk.
Definition 2.1. Let φ ∈ H ∗ and ξ ∈ H. We define φ ⊗ ξ ∈ B(H) by
(φ ⊗ ξ)η = φ(η)ξ, ∀ η ∈ H.
Theorem 2.2. Let H be a complex Hilbert space, B(H) the algebra of all
bounded linear operators on H. Let Ma,b : B(H) → B(H) be defined by
Ma,b (x) = axb, ∀ x ∈ B(H) where a, b are fixed in B(H). Then kMa,b xk =
kakkbk.
Proof. By definition, kMa,b |B(H)k = sup {kMa,b (x)k : x ∈ B(H), kxk = 1} .
This implies that kMa,b |B(H)k ≥ kMa,b (x)k, ∀ x ∈ B(H), kxk = 1.
So ∀ ² > 0, kMa, b |B(H)k − ² < kMa,b (x)k, ∀ x ∈ B(H), kxk = 1.
But, kMa,b |B(H)k − ² < kaxbk ≤ kakkxkkbk = kakkbk.
Since ² is arbitrary, this implies that
kMa,b |B(H)k ≤ kakkbk.
On the other hand, let ξ, η ∈ H, kξk = kηk = 1, φ ∈ H ∗ .
Now,
kMa,b |B(H)k ≥ kMa,b (x)k, ∀ x ∈ B(H), kxk = 1.
(1)
A two-sided multiplication operator norm
21
But,
kMa,b (x)k = sup {k(Ma,b (x))ηk : ∀ η ∈ H, kηk = 1}
= sup {k(axb)ηk : η ∈ H, kηk = 1} .
Setting a = (φ ⊗ ξ1 ), ∀ ξ1 ∈ H, kξ1 k = 1 and
b = (ϕ ⊗ ξ2 ), ∀ ξ2 ∈ H, kξ2 k = 1, we have,
kMa,b |B(H)k ≥
=
=
=
=
=
=
=
=
kMa,b (x)k ≥ k(Ma,b (x))ηk
k(axb)ηk
k((φ ⊗ ξ1 )x(ϕ ⊗ ξ2 ))ηk
k(φ ⊗ ξ1 )x(ϕ(η)ξ2 )k
k(φ ⊗ ξ1 )ϕ(η)x(ξ2 )k
|ϕ(η)|k(φ ⊗ ξ1 )x(ξ2 )k
|ϕ(η)|kφ(x(ξ2 ))ξ1 k
|ϕ(η)||φ(x(ξ2 ))|kξ1 k
kakkbk.
Therefore,
kMa,b |B(H)k ≥ kakkbk.
(2)
Hence by inequalities (1)and (2),
kMa,b |B(H)k = kakkbk.
This completes the proof.
3
The Jordan Elementary Operator
Theorem 3.1. Let H be a 2-dimensional complex Hilbert space, B(H) the
algebra of bounded linear operators on H. Let Ta,b : B(H) → B(H) be defined
by Ta,b (x) = axb + bxa, ∀ x ∈ B(H) where a, b are fixed in B(H) and {e1 , e2 }
an orthonormal basis for H. Then for a constant C > 0 such that kTa,b k ≥
Ckakkbk, C = 1.
Proof. The proof of this theorem follows immediately from the results obtained
in [3].
Remark 3.2. From [13], we see that C = 1 is also true for symmetric operators (in this case, a and b are self adjoint).
Theorem 3.3. Let a, b ∈ Symm(H). Then kUa,b |Ak ≥ kakkbk.
See [13] for proof.
22
N. B. Okelo and J. O. Agure
Acknowledgements:
We thank the referees and reviewers for their useful comments and suggestions.
References
[1] M. Barraa and M. Boumazgour, A lower bound of the Norm of Operator
X → AXB + BXA, Extracta Math., 16(2001), 223-227.
[2] M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl., 342(2008), 386-393.
[3] A. Blanco, M. Boumazgour and T. J Ransford, On the norm of elementary
operators, J. London Math. Soc., 70(2004), 479-498.
[4] M. Cabrera and A. Rodriguez, Nondegenerately ultraprime Jordan Banach algebras, Proc. London Math. Soc., 69(1994), 576-604.
[5] B. Magajna, The norm of a symmetric elementary operator, Proc. Amer.
Math. Soc., 132(2004), 1747-1754.
[6] M. Mathieu, More properties of the product of two derivations of a C*algebra, Canad. Math. Bull., 42(1990), 115-120.
[7] M. Mathieu, Elementary operators on Calkin algebras, Irish Math. Soc.
Bull., 46(2001), 33-42.
[8] F. O. Nyamwala and J. O. Agure, Norms of Elementary Operators in
Banach Algebras, Int. Journal of Math. Analysis., Vol.2(9)(2008), 411424.
[9] A. Seddik, On the numerical range and norm of elementary operators,
Linear Multilinear Algebra., 52(2004), 293-302.
[10] A. Seddik, On the norm of elementary operators in standard operator
algebras, Acta Sci. Math. (Szeged), 70(2004), 229-236.
[11] A. Seddik, Rank one operators and norm of elementary operators, Linear
Algebra and its Applications, 424(2007), 177-183.
[12] L. L. Stacho and B. Zalar, On the norm of Jordan elementary operators
in standard operator algebras, Publ. Math. Debrecen, 49(1996), 127-134.
[13] L. L. Stacho and B. Zalar, Uniform primeness of Jordan algebra of symmetric elementary operators, Proc. Amer. Math. Soc., 126(1998), 22412247.
A two-sided multiplication operator norm
23
[14] J. G Stampfli, The norm of a derivation, Pacific journal of mathematics,
Vol.33(3)(1970), 737-747.
[15] R. M. Timoney, Norms of elementary operators, Irish Math. Soc. Bulletin,
46(2001), 13-17.
[16] R. M. Timoney, Norms and CB norms of Jordan elementary operators,
Bull. Sci. Math., 127(2003), 597-609.
[17] R. M. Timoney, Computing the norm of elementary operators, Illinois J.
Math., 47(2003), 1207-1226.
[18] R. M. Timoney, Some formulae for norms of elementary operators, J.
Operator Theory, 57(2007), 121-145.